What's is the restrictions for the function MatrixExp?

Question

When I use the MatrixExp on a general $2\times2$ matrix, Mathematica gives me this result:

MatrixExp[{{a,b},{c,d}}] // TraditionalForm

$\frac{1}{2\triangle}\left(\begin{array}{cc} e^{\frac{\triangle+a+d}{2}}\left(\triangle+a-d\right)-e^{\frac{-\triangle+a+d}{2}}\left(-\triangle+a-d\right) & 2be^{\frac{\triangle+a+d}{2}}-2be^{\frac{-\triangle+a+d}{2}}\\ 2ce^{\frac{\triangle+a+d}{2}}-2ce^{\frac{-\triangle+a+d}{2}} & e^{\frac{\triangle+a+d}{2}}\left(\triangle-a+d\right)-e^{\frac{-\triangle+a+d}{2}}\left(-\triangle-a+d\right) \end{array}\right)$

where $\triangle=\sqrt{a^{2}-2ad+4bc+d^{2}}$

Is this true for any matrix {{a,b},{c,d}} (including complex matrix)? If it is not, what is the restrictions on the matrix {{a,b},{c,d}}?

Answer 1

By default, all variables in Mathematica are assumed complex, and generic results are calculated if you don't specify any restrictions, e.g., with Assumptions.

So the above result is generically true for complex matrices. But it's not true for non-generic matrices, where some elements have special values. For example, take

MatrixExp[{{a,b},{c,d}}]/.{c->0,d->a}

which gives errors. You would have to calculate such specific cases separately:

MatrixExp[{{a, b}, {0, a}}]

(* ==> {{E^a, b E^a}, {0, E^a}} *)